There may be occasions where you assign tasks to a generic active directory account rather than to one specific person so that anyone within that department can deal with the request.
But for this to function as desired you will need to set the ‘TaskProcessOwner” property of the task process in the workflow generating the task. To do this you should right click on the line that generates the task and then set the ‘TaskProcessOwner” property. You may select an individual user or a SharePoint group. This user or group will then be able to complete tasks on behalf of the assignee.